Order Statistic question

Question

I guess the condition should be $E[2Y_3]=\theta$ ? Am I right ? And I appplied first $$f_{(3)}(x)=nf(x)\binom{n-1}{3-1}F^{2}(x)(1-F(x))^{n-3}$$

But can you explain the further procedure and $F(x)$ also ??

Give me more hints or help me in solving this :)

Answer 1

Consider $z=x-\frac\theta2$. By symmetry about the origin, the expectation of $Z_{(3)}=Y_{(3)}-\frac\theta2$ must vanish.

Answer 2

Given a sample of size $n$ and $1\le k\le n$,

$$ \mathsf{E}Y_{(k)}=t\times\frac{n!}{(k-1)!(n-k)!}\int_0^1 u^k(1-u)^{n-k}\,du=t\times \frac{k}{n+1}. $$